zbMATH — the first resource for mathematics

Maximal injective subalgebras in factors associated with free groups. (English) Zbl 0545.46041
In this paper concrete examples of maximal injective von Neumann algebras in type \(II_ 1\) factors are constructed. In particular if the free group of n generators \({\mathbb{F}}_ n\) acts freely on some nonatomic probability space \((X,\mu)\) by measure preserving automorphisms and M denotes the associated group measure algebra and \(R_ u\) denotes the injective subalgebra of M corresponding to the action of the generator \(u\in {\mathbb{F}}_ n\) on \((X,\mu)\), then \(R_ u\) is a maximal injective von Neumann subalgebra of M. The subalgebra \(R_ u\) can be any injective type \(II_ 1\) von Neumann algebra provided with a suitable action of \({\mathbb{F}}_ n\) on \((X,\mu)\).
Two old problems of R. V. Kadison on the embedding of the hyperfinite factor \(R_ u\) posed in the Baton Rouge Conference are also solved.
Reviewer: V.I.Ovchinnikov

46L10 General theory of von Neumann algebras
46L35 Classifications of \(C^*\)-algebras
22D15 Group algebras of locally compact groups
46L55 Noncommutative dynamical systems
Full Text: DOI
[1] Ching, W.M, Free products of von Neumann algebras, Trans. amer. math. soc., 178, 147-163, (1973) · Zbl 0264.46066
[2] Choda, M, The crossed product of a full II_{1}-factor by a group of outer automorphisms, Math. japon., 23, 4, 385-391, (1978) · Zbl 0395.46047
[3] {\scM. Choda}, Inner amenability and fullness, to appear. · Zbl 0537.46052
[4] Connes, A, Classification of injective factors, Ann. of math., 104, 73-116, (1976) · Zbl 0343.46042
[5] Connes, A, Outer conjugacy classes of automorphisms of factors, Ann. ecole norm. sup., 8, 383-419, (1975) · Zbl 0342.46052
[6] Connes, A; Feldman, J; Weiss, B, An amenable equivalence relation is generated by a single transformation, Ergod. theory and dynam. systems, 1, 431-450, (1981) · Zbl 0491.28018
[7] Connes, A; Störmer, E, Entropy of automorphisms of II_{1} von Neumann algebras, Acta math., 134, 289-306, (1975) · Zbl 0326.46032
[8] Dixmier, J, Quelques propriétés des suites centrales dans LES facteurs de type II_{1}, Invent. math., 7, 215-225, (1969) · Zbl 0174.18702
[9] Dye, H; Dye, H, On groups of measure preserving transformations, I, II, Amer. J. math., Amer. J. math., 85, 551-576, (1963) · Zbl 0191.42803
[10] Fuglede, B; Kadison, R.V, On a conjecture of murray and von Neumann, (), 420-425 · Zbl 0043.11702
[11] Jones, V.F.R, A converse to Ocneanu’s theorem, J. oper. theory, 10, (1983), in press · Zbl 0547.46045
[12] Jones, V.F.R, Central sequences in crossed products of full factors, Duke math. J., 49, 29-34, (1982) · Zbl 0492.46049
[13] Kadison, R.V, Normalcy in operator algebras, Duke math. J., 29, 459-464, (1962) · Zbl 0177.17802
[14] {\scR. V. Kadison}, Problems on von Neumann algebras, Baton Rouge Conference, unpublished.
[15] Lyndon, R; Schupp, P, Combinatorial group theory, (1977), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0368.20023
[16] McDuff, D, Central sequences and the hyperfinite factor, (), 443-461 · Zbl 0204.14902
[17] Murray, F; von Neumann, J, Rings of operators, IV, Ann. of math., 44, 716-808, (1943) · Zbl 0060.26903
[18] {\scA. Ocneanu}, “Actions of Discrete Amenable Groups on Factors,” Thesis, Warwick University. · Zbl 0608.46035
[19] Philips, J, Automorphisms of full II_{1}-factors with applications to factors of type III, Duke math. J., 43, 375-385, (1976) · Zbl 0329.46061
[20] Popa, S, On a problem of R. V. kadison on maximal abelian ∗-subalgebras in factors, Invent. math., 65, 269-281, (1981) · Zbl 0481.46028
[21] Popa, S, Orthogonal pairs of subalgebras in finite von Neumann algebras, J. oper. theory, 9, 253-268, (1983) · Zbl 0521.46048
[22] Schmidt, K, Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group actions, Ergod. theory and dynam. systems, 1, 223-236, (1981) · Zbl 0485.28019
[23] Strǎtilǎ, S, Modular theory in operator algebras, (1981), Editura Academiei/Abacus Press Bucuresti/Tunbridge Wells · Zbl 0504.46043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.